248
7 Mathematical Models of Functionally Graded Beams in Temperature Field
investigated by Li et al. [142]. Employing Euler-Bernoulli beam theory, geometrically nonlinear dynamic governing equations (PDEs) have been derived taking
into account the thermo-electromechanical loadings, and then the problem has been
reduced to two sets of coupled ODEs. Thermoelastic post-buckling equilibrium paths
and characteristic curves of the first three natural frequencies versus the temperature, the electricity and the gradient parameters have been reported. In particular, it
has been illustrated how the tensional force produced in the piezoelectric layers can
increase the critical buckling temperature/natural frequency.
Kiani and Eslami [153] have investigated thermo-mechanical buckling of
temperature-dependent FGM beams. They have used the Timoshenko model in which
properties of the constituents depended on the temperature and thickness. Various
types of boundary conditions have been investigated and closed forms of the solutions
for the critical buckling temperature of the beams have been presented.
The model of gold nanobeam vibrations induced by laser pulse heating has been
derived by Youssef et al. [154] in the context of two-temperature generalized thermoelasticity and non-Fourier heat conduction. It has been shown that an increase
in the value of the two-temperature parameter yields a decrease in the values of the
stress-strain energy, and the latter process has been damped.
Prabhakar and Vengallatore [155] have studied the effects of beam geometry, natural frequency, flexural mode shapes and structural boundary conditions on thermoelastic damping of single-crystal silicon microbeam resonators. A Green’s function
method has been applied to solve the 2D heat conductions equation, and a formula
for thermoelastic damping has been derived in the form of infinite series.
Free vibration characteristic of an axially loaded FGM cantilever Euler-Bernoulli
beam subjected to temperature rising have been studied by Akbas [156], where beam
material properties have been assumed to be temperature-dependent and changed
according to a low-power function.
Esfahani et al. [157] have investigated vibrations of an FGM beam under inplane thermal loading in the pre-buckling and post-buckling regimes, taking into
account material properties versus both position and temperature. It has been shown
that, depending on the type of boundary conditions and loading, free vibrations of
the beam under in-plane thermal loading may achieve zero at a certain temperature
indicating existence of a bifurcation-type instability.
The research results reported in this chapter extend many earlier investigations
on thermal field influence on the thermodynamics of structural members including
Euler-Bernoulli beams [118, 119, 158–161].
7.6.2 Mathematical Model
The studied beam consists of a curvilinear body of the length l, height h and curvature
k x = 1/R x (see Fig. 7.15).
The mathematical model of the beam is introduced based on the following assumptions:
7 Mathematical Models of Functionally Graded Beams in Temperature Field
investigated by Li et al. [142]. Employing Euler-Bernoulli beam theory, geometrically nonlinear dynamic governing equations (PDEs) have been derived taking
into account the thermo-electromechanical loadings, and then the problem has been
reduced to two sets of coupled ODEs. Thermoelastic post-buckling equilibrium paths
and characteristic curves of the first three natural frequencies versus the temperature, the electricity and the gradient parameters have been reported. In particular, it
has been illustrated how the tensional force produced in the piezoelectric layers can
increase the critical buckling temperature/natural frequency.
Kiani and Eslami [153] have investigated thermo-mechanical buckling of
temperature-dependent FGM beams. They have used the Timoshenko model in which
properties of the constituents depended on the temperature and thickness. Various
types of boundary conditions have been investigated and closed forms of the solutions
for the critical buckling temperature of the beams have been presented.
The model of gold nanobeam vibrations induced by laser pulse heating has been
derived by Youssef et al. [154] in the context of two-temperature generalized thermoelasticity and non-Fourier heat conduction. It has been shown that an increase
in the value of the two-temperature parameter yields a decrease in the values of the
stress-strain energy, and the latter process has been damped.
Prabhakar and Vengallatore [155] have studied the effects of beam geometry, natural frequency, flexural mode shapes and structural boundary conditions on thermoelastic damping of single-crystal silicon microbeam resonators. A Green’s function
method has been applied to solve the 2D heat conductions equation, and a formula
for thermoelastic damping has been derived in the form of infinite series.
Free vibration characteristic of an axially loaded FGM cantilever Euler-Bernoulli
beam subjected to temperature rising have been studied by Akbas [156], where beam
material properties have been assumed to be temperature-dependent and changed
according to a low-power function.
Esfahani et al. [157] have investigated vibrations of an FGM beam under inplane thermal loading in the pre-buckling and post-buckling regimes, taking into
account material properties versus both position and temperature. It has been shown
that, depending on the type of boundary conditions and loading, free vibrations of
the beam under in-plane thermal loading may achieve zero at a certain temperature
indicating existence of a bifurcation-type instability.
The research results reported in this chapter extend many earlier investigations
on thermal field influence on the thermodynamics of structural members including
Euler-Bernoulli beams [118, 119, 158–161].
7.6.2 Mathematical Model
The studied beam consists of a curvilinear body of the length l, height h and curvature
k x = 1/R x (see Fig. 7.15).
The mathematical model of the beam is introduced based on the following assumptions: